Singular Integro-differential Operators Research Articles

On a New Class of Singular Integro-differential Equations

In this paper for a new class of model and non-model partial integro-differential equations with singularity in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions. Such type of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of analytic functions. In the process of our research the new types of singular integro-differential operators are introduced and main property of entered operators are learned. It is shown that the solution of studied equation is equivalent to the solution of system of two equations with respect to x and y, one of which is integral equation and the other is integro-differential equation. Further, non-model integro-differential equations are studied by regularization method. This regularization method for non-model equation is based on selecting and analysis of a model part of the equation and reduced to the solution of two second kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence of a non-model part in the equation does not affect to the general structure of the solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important. In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a Cauchy type problems are investigated.

Linear stability of inviscid vortex rings to axisymmetric perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (J. Fluid Mech., vol. 57, 1973, pp. 417–431). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ‘fat’. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine’s vortex and Hill’s vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.

Linear stability of Hill’s vortex to axisymmetric perturbations

We consider the linear stability of Hill’s vortex with respect to axisymmetric perturbations. Given that Hill’s vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretized with a highly accurate spectral approach. This operator has two unstable and two stable eigenvalues complemented by a continuous spectrum of neutrally stable eigenvalues. By considering a family of suitably regularized (smoothed) eigenvalue problems solved with a range of numerical resolutions, we demonstrate that the corresponding eigenfunctions are in fact singular objects in the form of infinitely sharp peaks localized at the front and rear stagnation points. These findings thus refine the results of the classical analysis by Moffatt & Moore (J. Fluid Mech., vol. 87, 1978, pp. 749–760).

Transmutation operators associated with an integro-differential operator on the real line and certain of their applications

We consider a singular integro-differential operator Λ on the real line. We build transmutation operators of Λ and its dual Λ˜ into the first derivative operator d/dx. Using these transmutation operators, we develop a new commutative harmonic analysis on the real line corresponding to the operator Λ.

Intertwining Operators Associated with a Singular Integro-Differential Operator on the Real Line and Certain of Their Applications

We consider a singular integro-differential operator Λ on the real line which includes as a particular case the Dunkl operator associated with the reflection group Z2 on R. We build intertwining operators of Λ and its dual Λ into the first derivative operator d/dx. Using these intertwining operators, we firstly establish a PaleyWiener theorem for the Fourier transform associated to Λ, and secondly introduce a generalized convolution on R tied to Λ.

Spectral asymptotics and the regularized trace of a singular integral operator

We give more precise asymptotic behaviour of the spectrum of a singular integro-differential operator and find the regularized trace of its inverse operator.Bibliography: 12 titles.

Identification problems for nonclassical integro-differential parabolic equations

In this paper we identify a convolution kernel k — i.e. we prove existence and uniqueness of k — in a first-order singular integrodifferential operator equation of Volterra type in two overdetermined problems in the framework of Hilbert spaces. We stress that in the first problem, by virtue of specific nonlocal time conditions, the kernel can be recovered globally in time, while in the latter only locally in time because of conditions that are of time-periodic type.

Identification problems for nonclassical integro-differential parabolic equations

In this paper we identify a convolution kernel k — i.e. we prove existence and uniqueness of k — in a first-order singular integrodifferential operator equation of Volterra type in two overdetermined problems in the framework of Hilbert spaces. We stress that in the first problem, by virtue of specific nonlocal time conditions, the kernel can be recovered globally in time , while in the latter only locally in time because of conditions that are of time-periodic type.

On Singular Integro‐Differential Operators on the Halfaxis

AbstractIn the frame of classical SOBOLEV spaces the FREDHOLM property will be characterized for a class of singular integro‐differential operators on the positive halfaxis. A symbol calculus moreover permits to specify an index formula and a special FREDHOLM inverse.

Gel'Fand Theory of Pseudo Differential Operators

Introduction. The conventional pseudo differential operators were introduced by Kohn and Nirenberg [8] and extensively studied in a generalized version by Hdrmander [7]. In either case they appear as operators acting on C, -functions of a differentiable manifold; their definition is designed to obtain a class of linear operators containing all linear partial differential operators with coefficients behaving like constants at infinity, and having properties like such differential operators. Many other important classes of linear operators are also special classes of pseudo differential operators over Rn: a) singular integral operators with C' -kernels; b) singular integro-differential operators of the form 7 KYDa IaI _N studied earlier by Dynin [4] and Seeley [10], with coefficients Ka being singular integral operators; c) regular and singular convolution operators, especially convolutions with functions having a singularity like I x 1-t at zero, as investigated earlier by Zarantonello [14]; d) integral operators with kernels like Green's functions. The Kohn-Nirenberg pseudo differential operators over Rn form an algebra, and the relation between this algebra and the algebra of partial differential operators over Rn might be compared with the relation between a quotient field and its generating ring. The various resulting advantages are essentially derived from the fact that a pseudo differential operator can be inverted within the algebra, if at all. All pesudo differential operators have an order, defined as a real number t such that the operator possesses extensions which map every L2-Soboleff space 1 continuously into >'~t. An operator of all orders is said to have order -oo. Also to every pseudo differential operator over Rn there is asso-

Non-Local Elliptic Boundary-Value Problems

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.